Dispensing Coffee. A coffee machine is supposed to dispense 8 fluid ounces (floz) of coffee into a paper cup. In reality, the amounts dispensed vary from cup to cup. However, if the machine is working properly, most of the cups will contain within 15% of the advertised 8 fl oz. In other words, the standard deviation of the amounts dispensed should be less than 0.4 fl oz. A random sample of 15 cups provided the following data, in fluid ounces.

7.24 7.79 8.20 8.09 7.92 8.02 7.64 7.89 8.18 7.80 8.00 7.80 8.14 8.04 8.18

a. At the 5% significance level, do the data provide sufficient evidence to conclude that the standard deviation of the amounts being dispensed is less than 0.4 fl oz? (Note: s = 0.255.)

b. Why is it important that the standard deviation of the amounts of coffee being dispensed not be too large?